Which Expression Is Equivalent To $9^{-2}$?

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Introduction

When dealing with exponents, it's essential to understand the properties and rules that govern them. One of the fundamental rules is the power of a power rule, which states that for any numbers a, b, and c, (a^b)c = a^(b*c). This rule can be applied to simplify expressions involving exponents. In this article, we will explore which expression is equivalent to $9^{-2}$.

Understanding Negative Exponents

Before we dive into the solution, it's crucial to understand what negative exponents represent. A negative exponent is a shorthand way of writing a fraction. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$. This means that $9^{-2}$ can be rewritten as $\frac{1}{9^2}$.

Simplifying the Expression

Now that we have rewritten $9^{-2}$ as $\frac{1}{9^2}$, we can simplify the expression further. To do this, we need to calculate the value of $9^2$. Since $9^2$ is equal to $9 \times 9$, we can multiply the numbers together to get $81$. Therefore, $\frac{1}{9^2}$ is equivalent to $\frac{1}{81}$.

Alternative Expressions

In addition to $\frac{1}{81}$, there are other expressions that are equivalent to $9^{-2}$. One of these expressions is $\frac{1}{(9^2)}$. This expression is equivalent to $\frac{1}{81}$, as we calculated earlier.

Conclusion

In conclusion, the expression equivalent to $9^{-2}$ is $\frac{1}{81}$. This can also be written as $\frac{1}{(9^2)}$. Understanding the properties of exponents and how to simplify expressions involving them is crucial in mathematics. By applying the power of a power rule and rewriting negative exponents as fractions, we can simplify complex expressions and arrive at the correct solution.

Additional Examples

To further illustrate the concept of equivalent expressions, let's consider a few more examples.

Example 1

Which expression is equivalent to $2^{-3}$?

Step 1: Rewrite the negative exponent as a fraction

$2^{-3}$ can be rewritten as $\frac{1}{2^3}$.

Step 2: Calculate the value of $2^3$

$2^3$ is equal to $2 \times 2 \times 2$, which is $8$.

Step 3: Simplify the expression

$\frac{1}{2^3}$ is equivalent to $\frac{1}{8}$.

Example 2

Which expression is equivalent to $5^{-2}$?

Step 1: Rewrite the negative exponent as a fraction

$5^{-2}$ can be rewritten as $\frac{1}{5^2}$.

Step 2: Calculate the value of $5^2$

$5^2$ is equal to $5 \times 5$, which is $25$.

Step 3: Simplify the expression

$\frac{1}{5^2}$ is equivalent to $\frac{1}{25}$.

Final Thoughts

In conclusion, understanding the properties of exponents and how to simplify expressions involving them is crucial in mathematics. By applying the power of a power rule and rewriting negative exponents as fractions, we can simplify complex expressions and arrive at the correct solution. The examples provided in this article demonstrate how to apply these concepts to real-world problems.

Frequently Asked Questions

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents a repeated multiplication of a number, while a negative exponent represents a repeated division of a number.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, you can use the following formula: $a^{-n} = \frac{1}{a^n}$.

Q: Can you provide more examples of equivalent expressions?

A: Yes, here are a few more examples:

• $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$
• $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• $6^{-3} = \frac{1}{6^3} = \frac{1}{216}$

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Negative Exponents" by Khan Academy
• [3] "Equivalent Expressions" by IXL Math

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

Introduction

Exponents and equivalent expressions are fundamental concepts in mathematics that can be challenging to understand, especially for students who are new to the subject. In this article, we will address some of the most frequently asked questions about exponents and equivalent expressions, providing clear and concise answers to help you better understand these concepts.

Q: What is an exponent?

A: An exponent is a small number that is written to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, in the expression $2^3$, the exponent $3$ indicates that the base number $2$ should be multiplied by itself three times: $2 \times 2 \times 2 = 8$.

Q: What is a negative exponent?

A: A negative exponent is a shorthand way of writing a fraction. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$. This means that $9^{-2}$ can be rewritten as $\frac{1}{9^2}$.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, you can use the following formula: $a^{-n} = \frac{1}{a^n}$. For example, $3^{-4}$ can be rewritten as $\frac{1}{3^4}$.

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents a repeated multiplication of a number, while a negative exponent represents a repeated division of a number. For example, $2^3$ represents $2 \times 2 \times 2$, while $2^{-3}$ represents $\frac{1}{2 \times 2 \times 2}$.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite the negative exponent as a fraction and then simplify the expression. For example, $9^{-2}$ can be rewritten as $\frac{1}{9^2}$, which is equivalent to $\frac{1}{81}$.

Q: Can you provide more examples of equivalent expressions?

A: Yes, here are a few more examples:

• $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$
• $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• $6^{-3} = \frac{1}{6^3} = \frac{1}{216}$

Q: How do I apply the power of a power rule?

A: The power of a power rule states that for any numbers a, b, and c, (a^b)c = a^(b*c). This means that you can simplify an expression by multiplying the exponents together. For example, $(2^3)^4 = 2^(3*4) = 2^12$.

Q: Can you provide more examples of the power of a power rule?

A: Yes, here are a few more examples:

• $(3^2)^3 = 3^(2*3) = 3^6$
• $(4^4)^2 = 4^(4*2) = 4^8$
• $(5^3)^5 = 5^(3*5) = 5^15$

Q: How do I apply the product of powers rule?

A: The product of powers rule states that for any numbers a, b, and c, a^b * a^c = a^(b+c). This means that you can simplify an expression by adding the exponents together. For example, $2^3 * 2^4 = 2^(3+4) = 2^7$.

Q: Can you provide more examples of the product of powers rule?

A: Yes, here are a few more examples:

• $3^2 * 3^3 = 3^(2+3) = 3^5$
• $4^4 * 4^5 = 4^(4+5) = 4^9$
• $5^3 * 5^4 = 5^(3+4) = 5^7$

Q: How do I apply the quotient of powers rule?

A: The quotient of powers rule states that for any numbers a, b, and c, a^b / a^c = a^(b-c). This means that you can simplify an expression by subtracting the exponents. For example, $2^3 / 2^2 = 2^(3-2) = 2^1$.

Q: Can you provide more examples of the quotient of powers rule?

A: Yes, here are a few more examples:

• $3^2 / 3^1 = 3^(2-1) = 3^1$
• $4^4 / 4^3 = 4^(4-3) = 4^1$
• $5^3 / 5^2 = 5^(3-2) = 5^1$

Final Thoughts

In conclusion, understanding exponents and equivalent expressions is crucial in mathematics. By applying the power of a power rule, product of powers rule, and quotient of powers rule, you can simplify complex expressions and arrive at the correct solution. We hope that this article has provided you with a better understanding of these concepts and has helped you to become more confident in your math skills.

Frequently Asked Questions

Q: What is the difference between a positive and negative exponent?

A: A positive exponent represents a repeated multiplication of a number, while a negative exponent represents a repeated division of a number.

Q: How do I rewrite a negative exponent as a fraction?

A: To rewrite a negative exponent as a fraction, you can use the following formula: $a^{-n} = \frac{1}{a^n}$.

Q: Can you provide more examples of equivalent expressions?

A: Yes, here are a few more examples:

• $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$
• $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• $6^{-3} = \frac{1}{6^3} = \frac{1}{216}$

References

• [1] "Exponents and Exponential Functions" by Math Open Reference
• [2] "Negative Exponents" by Khan Academy
• [3] "Equivalent Expressions" by IXL Math

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.